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Giải bài 10 trang 73 SGK Toán 10 tập 1 – Chân trời sáng tạo

Cho tứ giác lồi ABCD có các đường chéo AC = x,BD = y và góc giữa AC và BD bằng alpha Gọi S là diện tích của tứ giác ABCD.

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Đề bài

Cho tứ giác lồi ABCD có các đường chéo \(AC = x,BD = y\) và góc giữa AC và BD bằng \(\alpha .\) Gọi S là diện tích của tứ giác ABCD. a) Chứng minh \(S = \frac{1}{2}xy.\sin \alpha \) b) Nêu kết quả trong trường hợp \(AC \bot BD.\)

Phương pháp giải - Xem chi tiết

a) Tính diện tích 4 tam giác nhỏ theo \(\sin \alpha \). Chú ý: \(\sin ({180^o} - \alpha ) = \sin \alpha \) b) \(\alpha  = {90^o}\) thì \(\sin \alpha  = 1\)

Lời giải chi tiết

Gọi O là giao điểm của AC và BD.a) Áp dụng công thức \(S = \frac{1}{2}ac.\sin B\), ta có:\(\begin{array}{l}{S_{OAD}} = \frac{1}{2}.OA.OD.\sin \alpha ;\quad {S_{OBC}} = \frac{1}{2}.OB.OC.\sin \alpha ;\\{S_{OAB}} = \frac{1}{2}.OA.OB.\sin ({180^o} - \alpha );\quad {S_{OCD}} = \frac{1}{2}.OD.OC.\sin ({180^o} - \alpha ).\end{array}\)Mà \(\sin ({180^o} - \alpha ) = \sin \alpha \)\( \Rightarrow {S_{OAB}} = \frac{1}{2}.OA.OB.\sin \alpha ;\quad {S_{OCD}} = \frac{1}{2}.OD.OC.\sin \alpha .\)\(\begin{array}{l} \Rightarrow {S_{ABCD}} = \left( {{S_{OAD}} + {S_{OAB}}} \right) + \left( {{S_{OBC}} + {S_{OCD}}} \right)\\ = \frac{1}{2}.OA.\sin \alpha .(OD + OB) + \frac{1}{2}.OC.\sin \alpha .(OB + OD)\\ = \frac{1}{2}.OA.\sin \alpha .BD + \frac{1}{2}.OC.\sin \alpha .BD\\ = \frac{1}{2}.BD.\sin \alpha .(OA + OC)\\ = \frac{1}{2}.AC.BD.\sin \alpha  = \frac{1}{2}.x.y.\sin \alpha .\end{array}\)b) Nếu \(AC \bot BD\) thì \(\alpha  = {90^o} \Rightarrow \sin \alpha  = 1.\)\( \Rightarrow {S_{ABCD}} = \frac{1}{2}.x.y.1 = \frac{1}{2}.x.y.\)

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